Debye Model For Specific Heat

In thermodynamics and solid state physics, the Debye model is a method developed by Peter Debye in 1912[7] for estimating the phonon contribution to the specific heat (heat capacity) in a solid[1]. This model correctly explains the low temperature dependence of the heat capacity, which is proportional to T3. It also recovers the Dulong-Petit law at high temperatures. But due to simplifying assumptions, its accuracy suffers at intermediate temperatures.

## 1 Introduction to Phonons

In 1912 Debye realized that, inconsistent with the Einstein model, low-energetic excitations of a solid material were not oscillations of a single atom[2], but collective modes propagating through the material. Such vibrations can be considered to be sound waves, and their propagation speed is the speed of sound in the material[3]. Moreover, these modes only accept energy in discreet amounts.

Quantum theory uses the concepts of phonons, which are “quasi-particles” with definite energies and directions of motion, to treat the vibrations. The concept of phonon is analogous with photons of the electromagnetic wave. The relations between the energy of a phonon ε, the angular frequency ω and the wave vector $\vec{q}$ are:

(1)
\begin{align} \varepsilon=\hbar\omega \end{align}
(2)
\begin{align} \omega=\upsilon_{s}|\vec{q}| \end{align}

where νs is the velocity of the sound wave.

As a kind of Bosons, phonons obey Bose–Einstein statistics. The expectation number of bosons in a state with energy E is[6]:

(3)
\begin{align} n_{(E)}=\frac{1}{e^{E/k_{B}T}-1}=\frac{1}{e^{\hbar\omega/k_{B}T}-1} \end{align}

where kB= 1.380 6504(24)×10−23J/K is the Boltzmann constant.

## 2 Debye frequency and Debye Temperature

Unlike electromagnetic radiation in a box, a phonon cannot have infinite frequency. Its frequency is bound by the medium of its propagation — the atomic lattice of the solid. If there are N primitive cells in the specimen, the total number of phonon modes are N. A cut-off frequency ωD, known as Debye frequency, is determined by the following[4]:
In the 3 dimensional reciprocal space(See "Real and reciprocal crystal lattices" by fangxiao), the volume for each allowed wave vector $\vec{q}$ is

(4)
\begin{align} (\frac{2\pi}{L})^{3}=\frac{8\pi^{3}}{V} \end{align}

Since there is a cut-off wave vector $q_{D}=\omega_{D}/\upsilon_{s}$, all the modes are confined within a sphere with radius qD. Thus number of modes (not number of phonons) should be

(5)
\begin{align} N=(\frac{4}{3}\pi{q_{D}}^{3})/({\frac{8\pi^{3}}{V}}) \end{align}

Or

(6)
\begin{align} q_{D}=(6\pi^{2}\frac{N}{V})^{\frac{1}{3}} \end{align}
(7)
\begin{align} \omega_{D}=\upsilon_{s}(6\pi^{2}\frac{N}{V})^{\frac{1}{3}} \end{align}

Debye temperature TD is defined as

(8)
\begin{align} T_{D}=\frac{\hbar\omega_{D}}{k_{B}}=\frac{\hbar\upsilon_{s}}{k_{B}}(6\pi^{2}\frac{N}{V})^{\frac{1}{3}} \end{align}

The significance of this physical term will be discussed in the 4th section. For elements in the same group, heavier atoms have lower Debye temperatures, simply because the velocity of sound decreases as the density increases.[4] The Debye temperatures of several substances are listed in Table 1.[1]

Table1: Debye Temperatures of several substances.

 Aluminum 428K Cadmium 209K Chromium 630K Copper 343.5K Gold 165K
 Iron 470K Lead 105K Manganese 410K Nickel 450K Platinum 240K
 Silicon 645K Silver 225K Tantalum 240K Tin(white) 200K Titanium 420K
 Tungsten 400K Zinc 327K Carbon 2230K Ice 192K

## 3 Derivation for Specific Heat

In the Debye approximation, the velocity of sound $\upsilon_{s}$ is taken as constant for each polarization type, as it would be for a classical elastic continuum. According to equation (7), the density of states(See "Density of States" by vincent) is:

(9)
\begin{align} D_{(\omega)} =\frac{dN}{d\omega}=\frac{V\omega^{2}}{2\pi^{2}\upsilon_{s}^{3}} \end{align}

Thus, thermal energy for each polarization type is given by[4]:

(10)
\begin{align} U=\int d\omega D(\omega) n(\omega)\hbar\omega=\int_{0}^{\omega_{D}}d\omega\frac{V\omega^{2}}{2\pi^{2}\upsilon_{s}^{3}}\frac{\hbar\omega}{e^{\hbar\omega/k_{B}T}-1} \end{align}

There are 3 polarization types, 1 longitude and 2 transverse. For brevity, we assume that phonon velocity is independent of polarization. Thus we multiply a factor 3 and use equation (7) to obtain the total energy of phonon:

(11)
\begin{align} U=\frac{3V\hbar}{2\pi^{2}\upsilon_{s}^{3}}\int_{0}^{\omega_{D}}d\omega\frac{\omega^{3}}{e^{\hbar\omega/k_{B}T}-1}=\frac{3Vk_{B}^{4}T^{4}}{2\pi^{2}\upsilon_{s}^{3}\hbar^{3}}\int_{0}^{x_{D}}dx\frac{x^{3}}{e^{x}-1}=9Nk_{B}T(\frac{T}{T_{D}})^{3}\int_{0}^{x_{D}}dx\frac{x^{3}}{e^{x}-1} \end{align}

where x≡ћω/kBT and xD≡TD/T.

The heat capacity is:

(12)
\begin{align} C_{V}=\frac{\partial U}{\partial T}=9Nk_{B}(\frac{T}{T_{D}})^{3}\int_{0}^{x_{D}}dx\frac{x^{4}e^{x}}{(e^{x}-1)^{2}} \end{align}

At the left of figure 1[3] below, the experimental results of specific heats of four substances are plotted as a function of temperature and they look very different. But if they are scaled to T/TD, they look very similar and are very close to the Debye theory.

Figure1: Specific Heats of Lead, Silver, Aluminum and Diamond

## 4 High and Low Temperature Limits

The integral in equation (12) cannot be evaluated in closed form. But the high and low temperature limits can be assessed.

#### 4.1 High Temperature Limit

For the high temperature case where T»TD, the value of x is very small throughout the range of the integral. This justifies using the approximation to the exponential ex = 1 + x and reduces equation (11) and (12) to

(13)
\begin{align} U=9Nk_{B}T(\frac{T}{T_{D}})^{3}\int_{0}^{x_{D}}x^{2}dx=3Nk_{B}T \end{align}
(14)
$$C_{V}=3Nk_{B}$$

which is the classical Dulong-Petit result.

When the temperature is above the Debye temperature, the heat capacity is very close to the classical value $3Nk_{B} T$. For temperatures below the Debye temperature, quantum effects become important and CV decreases to zero. Note that diamond, with a Debye temperature of 1860K, is a “quantum solid” at room temperature.[8]

#### 4.2 Low Temperature Limit

At very low temperature where T« TD, only long wavelength acoustic modes are thermally excited. These are just the modes that can be treated as elastic continuum with macroscopic elastic constants. The energy of those short wavelength modes are too high to be populated significantly at low temperatures. We may approximate xD≡TD/T to infinity and make use of the standard integral

(15)
\begin{align} \int_{0}^{\infty}dx\frac{x^{3}}{e^{x}-1}=\frac{\pi^{4}}{15} \end{align}

to obtain

(16)
\begin{align} U=\frac{3\pi^{4}Nk_{B}T^{4}}{5T_{D}^{3}} \end{align}
(17)
\begin{align} C_{V}=\frac{12\pi^{4}Nk_{B}T^{3}}{5T_{D}^{3}}\cong324Nk_{B}\frac{T^{3}}{T_{D}^{3}} \end{align}

For actual crystals, the temperatures at which the T3 approximation holds are quite low, even may be below TD/50. It is easy to understand (T/TD)3 with a simple argument. Only the modes with ћω<kBT will be excited to any appreciable extent at a low temperature T. Define ωT≡kT/ћ as the largest frequency excited at this temperature. In the reciprocal space, the fraction occupied by the excited states is (qT/qD)3 or (wT/wD)3=(T/TD)3.

## 5 Extension: Einstein-Debye Specific Heat

This T3 dependence of the specific heat at very low temperatures agrees with experiment for nonmetals. For metals the specific heat of highly mobile conduction electrons is approximated by Einstein Model, which is composed of single-frequency quantum harmonic oscillators. The temperature dependence of Einstein model is just T. It becomes significant at low temperatures and is combined with the above lattice specific heat in the Einstein-Debye specific heat[3].

(18)
\begin{align} C_{metal}=C_{electron}+C_{phonon}=\frac{\pi^{2}Nk^{2}}{2E_{f}}T+\frac{12\pi^{4}Nk_{B}}{5T_{D}^{3}}T^{3} \end{align}

Finally, experiments suggest that amorphous materials do not follow the Debye T3 law even at temperatures below 0.01TD[8]. There is more yet to be learned.

References
1. http://en.wikipedia.org/wiki/Debye_model
2. http://www.uam.es/personal_pdi/ciencias/mramos/Docencia_archivos/debye.pdf
3. http://hyperphysics.phy-astr.gsu.edu/Hbase/Solids/phonon.html
4. C. Kittel, introduction to solid state physics, pp.117-140. John Wiley & Sons, Inc., 1996
5. L. Landau and E. Lifshitz, Statistical Physics, Part, pp.191 -217. Elsevier, 1984
6. http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_statistics
7. Zur Theorie der spezifischen Waerm, Annalen der Physik (Leipzig) 39(4)
8. http://www.phys.unsw.edu.au/~gary/SM3_6.pdf